Theorem elz12s 28643

Description: Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
elz12s	⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elz12s

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3476	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V)
2		id 23	. . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 = (𝑥 /su (2s↑s𝑦)))
3		ovex 7445	. . . . 5 ⊢ (𝑥 /su (2s↑s𝑦)) ∈ V
4	2, 3	eqeltrdi 2871	. . . 4 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
5	4	a1i 11	. . 3 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℕ0s) → (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V))
6	5	rexlimivv 3207	. 2 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
7		eqeq1 2767	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝑥 /su (2s↑s𝑦))))
8	7	2rexbidv 3230	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
9		df-z12s 28586	. . 3 ⊢ ℤs[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦))}
10	8, 9	elab2g 3640	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
11	1, 6, 10	pm5.21nii 381	1 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455  (class class class)co 7412   /_su cdivs 28358  ℕ_0scn0s 28483  ℤ_sczs 28549  2_sc2s 28581  ↑_scexps 28583  ℤ_s[1/2]cz12s 28585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rex 3090  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-sn 4591  df-pr 4593  df-uni 4874  df-iota 6494  df-fv 6546  df-ov 7415  df-z12s 28586

This theorem is referenced by:  elz12si  28644  zz12s  28646  z12no  28647  z12addscl  28648  z12negscl  28649  z12shalf  28651  z12zsodd  28653  z12sge0  28654